Independent solution

How to solve this Duration and Convexity question

Setup

Setup

Keep the payment times at years two, three, and four and discount every amount at the given annual effective rate.

v=(1.07)−1v=(1.07)^{-1}

Model

Model

Macaulay duration divides the time-weighted present value by the ordinary present value.

DM=2(40000)v2+3(25000)v3+4(100000)v440000v2+25000v3+100000v4D_M=\frac{2(40000)v^2+3(25000)v^3+4(100000)v^4}{40000v^2+25000v^3+100000v^4}

Compute

Compute

The discounted-value denominator is about 115,330.49 and the time-weighted numerator is about 382,167.24.

DM=382167.24115330.49=3.314D_M=\frac{382167.24}{115330.49}=3.314

Answer

Answer

Their ratio is 3.314 years, which rounds to the value listed in choice D.

DM≈3.3 years(D)\boxed{D_M\approx3.3\text{ years}\quad\text{(D)}}